HMMT 二月 2005 · 代数 · 第 4 题

英文解析

4. If a, b, c > 0, what is the smallest possible value of + + ? (Note thatc a bb x c denotes the greatest integer less than or equal to x .)
   Solution: 4
   Since b x c > x − 1 for all x , we have that
   ⌊ ⌋ ⌊ ⌋ ⌊ ⌋
   a + b b + c c + a a + b b + c c + a + + > + + − 3
   c a b c a b
   ( ) ( )
   ( )
   a b b c c a = + + + + + − 3 .
   b a c b a c
   But by the AM-GM inequality, each of the first three terms in the last line is at least